Study of Lenvatinib in Patients With Advanced Cancer and Aberrations in FGF/FGFR Signaling

Inclusion Criteria: * Age \>= 18 years old * Pathologically confirmed advanced or metastatic malignancy characterized by one or more of the following: Subject is intolerant of standard therapy. Malignancy is refractory to standard therapy. Malignancy relapsed after standard therapy. Malignancy for which there is no standard therapy that improves survival by at least 3 months. * Evaluable tumor(s) with documented alteration(s) in FGF/FGFR-related gene(s). The FGF/FGFR aberration(s) can be identified at any point in the subject's cancer course. FGF/FGFR testing must have been performed in a CLIA-certified laboratory. Amplification(s) and/or mutation frequenc(ies) will be defined according to the standard of the test used. One example, Foundation OneTM, defines amplifications as ≥ 6 copies and base substitution mutations as present if there is ≥ 5% mutant allele frequency. * Subjects must meet the following laboratory requirements at screening (may be repeated): Adequate bone marrow function: absolute neutrophil count ≥ 1,500/mL; hemoglobin ≥ 8.5 g/dL, platelets ≥ 75,000/mL. Adequate liver function: transaminases (AST/ALT) and alkaline phosphatase ≤ 3 (≤ 5 X Upper Limit of Normal (ULN) in the setting of liver metastasis) x ULN; bilirubin ≤ 1.5 x ULN. Adequate renal function: creatinine clearance ≥ 40 mL/min (Cockcroft Gault). Adequate blood coagulation: international normalized ratio (INR) ≤ 2.3. Serum amylase and lipase ≤ 1.5 x ULN. * Adequately controlled blood pressure (BP): BP ≤ 150/90 mm Hg at screening (may be repeated and may be controlled with anti-hypertensive medication). * Adequate performance status (PS): Eastern Cooperative Oncology Group (ECOG) PS 0-2 * Women of childbearing potential must have a negative baseline blood pregnancy test. Women and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) for the duration of study and for at least 30 days after discontinuation of study drug (the half life of lenvatinib is about 28 hours in patients with cancer). * Subjects must be off other anti-tumor agents for at least 5 half lives of the agent or 4 weeks from the last day of treatment, whichever is shorter. Endocrine therapies (for example for breast or prostate cancer) and anti-Her2 therapies (for example, trastuzumab or lapatinib) are allowed to continue while on this study. * Subjects may not be receiving any other experimental agents or agents that are not FDA approved. * Ability to understand and willingness to sign a written consent document. Exclusion Criteria: * Pregnant or lactating women. * Subjects with known hypersensitivity to any of the components or metabolites of the drug product. * Subjects with FGFR mutations known to be inactivating. Mutations of unknown significance (based on most currently available information) will be allowed. * Subjects who have not recovered from toxicities as a result of prior anticancer therapy, except alopecia and infertility. Recovery is defined as \< Grade 2 severity per Common Terminology Criteria for Adverse Events Version 4.0 (CTCAE v4.0). * Significant cardiovascular impairment: history of congestive heart failure greater than New York Heart Association (NYHA) Class II, unstable angina, myocardial infarction or stroke within 6 months of the first dose of study drug, or cardiac arrhythmia requiring medical treatment at Screening. * Inability to swallow pills or determination by the investigator that absorption of oral medication would be impaired. * Major surgery (not including placement of central lines) within 3 weeks prior to randomization or planned surgery during the course of this study. * Any medical condition which, in the opinion of the investigator, would preclude study participation. * Subjects who are considered members of a vulnerable population (for example, prisoners).